Question 197: Oscillations

Question

All the springs in fig. (a), (b) and (c) are identical, each having force constant K each. Mass m is attached to each system. If $\mathrm{T}_a, \mathrm{~T}_b$ and $\mathrm{T}_{\mathrm{c}}$ are the time periods of oscillations of the three systems in fig. (a), (b) and (c) respectively, then

Accepted Answer

Finding the Time Periods
For any spring-mass system, the time period is $$ T = 2\pi \sqrt{\frac{m}{K'}} $$, where $m$ is the mass and $K'$ is the effective spring constant of the setup.
Figure (a):
There is only one spring with constant $K$. The time period is:
$$ T_a = 2\pi \sqrt{\frac{m}{K}} $$
Figure (b):
Two springs are attached in parallel. The effective spring constant adds up:
$ K' = K + K = 2K $
The time period is:
$$ T_b = 2\pi \sqrt{\frac{m}{2K}} $$
Simplify this:
$$ T_b = 2\pi \sqrt{\frac{m}{2K}} = 2\pi \frac{1}{\sqrt{2}} \sqrt{\frac{m}{K}} = \frac{T_a}{\sqrt{2}} $$
Figure (c):
Two springs are in series. Effective spring constant is:
$ \frac{1}{K'} = \frac{1}{K} + \frac{1}{K} = \frac{2}{K} $, so $ K' = \frac{K}{2} $
Time period is:
$$ T_c = 2\pi \sqrt{\frac{m}{K/2}} = 2\pi \sqrt{\frac{2m}{K}} = \sqrt{2} T_a $$
Shortest and Longest Time Period
$T_b$ (parallel springs) is the shortest, and $T_c$ (series springs) is the longest.
Relating the Time Periods
From above, we have:
$ T_a = \frac{T_c}{\sqrt{2}} $ and  $ T_a = \sqrt{2} T_b $
We can also write:
$ T_b = \frac{T_a}{\sqrt{2}} $ and $ T_c = \sqrt{2} T_a $
So, $ T_c $ is twice $ T_b $:
$ T_c = 2 T_b $

Answer

$\quad \mathrm{T}_{\mathrm{a}}=\sqrt{2} \mathrm{~T}_{\mathrm{b}}$

Answer

$\quad \mathrm{T}_{\mathrm{a}}=\frac{\mathrm{T}_{\mathrm{c}}}{\sqrt{2}}$

Answer

$\mathrm{T}_{\mathrm{b}}=2 \mathrm{~T}_{\mathrm{a}}$

Answer

$\quad \mathrm{T}_{\mathrm{b}}=2 \mathrm{~T}_{\mathrm{c}}$

Oscillations

All the springs in fig. (a), (b) and (c) are identical, each having force constant K each. Mass m is attached to each system. If \mathrm{T}_a, \mathrm{~T}_b and \mathrm{T}_{\mathrm{c}} are the time periods of oscillations of the three systems in fig. (a), (b) and (c) respectively, then

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